In this paper, a new notion of generalized super-weak Funk functions on a Finsler (or spray) space is introduced. We show that there is no non-trivial generalized super-weak Funk function on a complete spray space generalizing a result previously known only in the case of weak Funk functions on compact spray spaces. As its application, we prove that every complete Finsler space with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2160_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{S}\ge \lambda \tau ^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">S</mi> <mo>≥</mo> <mi>λ</mi> <msup> <mi>τ</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> for some positive constant <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2160_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> must be Riemannian where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2160_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\textbf{S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">S</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2160_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> are the S-curvature and the distortion respectively. We also give new or simple proofs of the Shen-Sun’s and Chen-Shen’s global rigidity theorems.

Some Rigidity Results on Complete Finsler (or spray) Spaces

期刊论文
发表日期：

2025年8月23日

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Hongru Song,
Xiaohuan Mo

摘要

In this paper, a new notion of generalized super-weak Funk functions on a Finsler (or spray) space is introduced. We show that there is no non-trivial generalized super-weak Funk function on a complete spray space generalizing a result previously known only in the case of weak Funk functions on compact spray spaces. As its application, we prove that every complete Finsler space with $\textbf{S}\ge \lambda \tau ^2$ $S \geq λ τ^{2}$ for some positive constant $\lambda $ $λ$ must be Riemannian where $\textbf{S}$ $S$ and $\tau $ $τ$ are the S-curvature and the distortion respectively. We also give new or simple proofs of the Shen-Sun’s and Chen-Shen’s global rigidity theorems.